Problem: Rewrite as a simplified fraction. ${1.8\overline{29} = {?}}$
Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 1829.2929...\\ 10x &= 18.2929...\end{align*}}$ Subtracting the two equations, we have: ${990x = 1811}$ ${x = \dfrac{1811}{990}} \text{or} {1 \dfrac{821}{990}}$